Q. No. 2Let N be the set of natural numbers and N* = N U {∞}. Define a4function d: N* × N* → R as follows; for m,n in N,d(m, п) %3DImd(m, 0) = d(∞, m) = = and d(0,00) = 0,mShow that (N",d) is a metric space.

I have proved the three conditions for metric

Q. No. 2 Let N be the set of natural numbers and N = N U {∞}. Define a function d: N* × N* → R as follows; for m,n in N, d(m, n) = Im d(m, 0) = d(∞,m) = - and d(∞, 00) = 0, 1 m Show that (N°, d) is a metric space.

Q. No. 2 Let N be the set of natural numbers and N = N U {∞}. Define a function d: N* × N* → R as follows; for m,n in N, d(m, n) = Im d(m, 0) = d(∞,m) = - and d(∞, 00) = 0, 1 m Show that (N°, d) is a metric space.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

(b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a Banach
ii).Now consider a function space of the set of functions f(x) of a real variable x E (-00, +o), such that f(x) → 0 as x - function w(x) = 1, determine if the following operators are Hermitian: i) id/dx , and ii) id5/dx5 t0o. For a unit weight
c) Let the set A = {-1,0, 1} and consider the following function: f3 A × A → A × A given by f(x,y) = (y7,x). Decide whether this function is injective, surjective, a bijection. f3 is an injection: Not Answered • ƒ3 is a surjection: Not Answered f3 is a bijection: Not Answered d) Let the set A = {1, 2, 3,4} and let P(A) denote the power-set of A. Consider the following function: f₁ : P(A) → P(A) given by ƒ(B) = {1, 2, 3}\B. Decide whether this function is injective, surjective, a bijection. f4 is an injection: Not Answered f4 is a surjection: Not Answered • f is a bijection: Not Answered
B. Show that the space V = {f: R → R} of functions from R to R is not finite- dimensional.
Let d: R2 × R2 →R given by xỉ + yỉ + x3 + y½ , if (x1,y1) # (x2, Y2) 0, if (x1, Y1) = (x2, Y2) d((x1.y1).(x2.y2))= { (a) Show that d is a metric. (b) Show that (R2, d) is a complete metric space. (c) Show that (R2, d no a connected metric space
Let Jz = {0, 1,2, 3, 4} and define a function g: J5 × Jz → Jz x J5 as follows: For all (a, b) e J, x J5, 9(а, b) 3D (5а — 3) тod 5, (4b + 2) тod 5). Find g(3, 4).
Calculate the Christoffel Symbols corresponding to ds² = (dx¹)² + G(x¹, x²) (dx²)², the metric. where G is a function of x¹ and x².
Let f(x,y)=g(x)h(y) where g and h are continuous functions on all real numbers such that g(x)=h(x+1) and R={(x,y) | 1 ≤ x ≤ 2, 2 ≤ y ≤3}. Consider the following statements (image)a. Only statement 2. is true b. All statements are false c. All statements are true d. Only 1 and 2 are true e. Only statement 3. is true f. Only 2 and 3 are true g. Only 1 and 3 are true h. Only statement 1. is true
#Showthat Don't copy
Let T be function from R to R de fined as T (x) = *2, where a (r1, 22) Then show that if T is linear operator or not
2. Let f: R→R be defined by f(x) = 4x -x2. Let B be the range of f. (a) Find B. (b) Fins f-¹([0, ∞ )). (c) Find a set ASR such that fl, the restriction of f on A, is a one-to-one and onto function from A to B, and find (fl)
Q1: Let M = R² be the set of all ordered pairs of real numbers and d: M x MR be a function defined as : d(x, y) = x₁=Y₁l+ 1x2 - y₂l, for all (x₁, x2), (₁,Y2) E M. Prove that the pair (M, d) forms a metric space.. Q2: write and prove the Additive property for the supremum.